Asymptotics for orthogonal polynomials and Christoffel functions on a ball
type: Item
instance of: scholarly article
title: Asymptotics for orthogonal polynomials and Christoffel functions on a ball
zbMATH Open document ID: 0888.33007
author: Yuan Xu
published in: Methods and Applications of Analysis
publication date: 1998-05-13T00:00:00Z
review text: The author investigates orthogonal polynomials in \(d\) real variables on the unit ball of \({\mathbf R}^d\) with respect to the weight function \(W_\mu({\mathbf x})=(1-|{\mathbf x}|)^{\mu-1/2}\). Let \({\mathbf P}_n({\mathbf x})\) be the vector of the orthonormal polynomials of degree \(n=n_1+n_2+\cdots+n_d\), then \({\mathbf P}_n({\mathbf x})\) has \({n+d-1 \choose n}\) components and the author has shown earlier that working with orthogonal polynomials in \(d\) variables is most convenient using this vector notation. This is particularly so in the present paper, where it is shown that for \(d \geq 2\) one can obtain the asymptotic behaviour of \({n+d-1 \choose n}^{-1} {\mathbf P}_n^T({\mathbf x}){\mathbf P}_n({\mathbf x})\) inside the open unit ball, and the limit is the ratio \(W_0({\mathbf x})/W_\mu({\mathbf x})\). Such a limit does not exist for \(d=1\), and in that case one usually investigates the asymptotic behaviour of the Christoffel function, which is \((\sum_{k=0}^n p_n^2(x))^{-1}\). For \(d \geq 2\) one can analyse each term in the Christoffel function separately, and hence, as a consequence the author also obtains the asymptotic behaviour of the Christoffel function, which was obtained earlier by \textit{L. Bos} [New Zealand J. Math. 23, 99-109 (1994; Zbl 0828.42014)]. The proof is based on an explicit formula for \({\mathbf P}_n^T({\mathbf x}){\mathbf P}_n({\mathbf x})\) in terms of an integral involving the Gegenbauer polynomial \(C_n^{\mu+(d-1)/2}\).
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 33C50
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 987700
zbMATH Keywords: orthogonal polynomials in several variables
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: Christoffel function
MaRDI profile type: Publication
full work available at URL: maa.1996.v3.n2.a6
OpenAlex ID: W124896198
DOI: 10.4310/MAA.1996.V3.N2.A6
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description: scientific article; zbMATH DE number 987700